A convenient choice of prior over transition probabilities is the Beta distribution. A Beta distribution, \(\mathrm {Beta}(\theta ; \alpha _0,\alpha _1)\) , over \(\theta \in [0, 1]\)  is determined by a pair of
shape parameters \(\alpha _0, \alpha _1\) . Different settings of the alphas can be used to encode distinct beliefs about \(\theta \) . As examples: \(\alpha _0 = \alpha _1 = 1\) 
is equivalent to a uniform distribution over \([0, 1]\) , \(\alpha _0 = \alpha _1 = 100\) 
is a unimodal distribution whose mode is at \(\theta = 0.5\) , and
\(\alpha _0 < 1, \alpha _1 < 1\)  (e.g., \(\alpha _0 = \alpha _1 = 0.5\) ) is a U-shaped distribution corresponding to the belief that \(\theta \) 
is likely to be close to 0 or 1. The Beta distribution is conjugate to the binomial
distribution [1]}, meaning that a
posterior that is the product of a Beta prior and a binomial
likelihood equals another Beta distribution whose parameters can be
calculated exactly.
